线性代数问题（关于方程组有解的条件）

Question

如题所述

Answer 1

方程组有解的充分必要条件是系数矩阵与增广矩阵有相同的秩
系数矩阵为

1
-1
0
0
0
1
-1
0
0
0

0
1
-1
0
0
0
1
-1
0
0

0
0
1
-1
0
=
0
0
1
-1
0
系数矩阵的秩为4

0
0
0
1
-1
0
0
0
1
-1

-1
0
0
0
1
0
0
0
0
0

增广矩阵为

1
-1
0
0
0
a1
1
-1
0
0
0
a1

0
1
-1
0
0
a2
0
1
-1
0
0
a2

0
0
1
-1
0
a3
=
0
0
1
-1
0
a3

0
0
0
1
-1
a4
0
0
0
1
-1
a4

-1
0
0
0
1
a5
0
0
0
0
0
a1+a2+a3+a4+a5
满足增广矩阵的秩为4，则a1+a2+a3+a4+a5=0